Although A/UX handles jobs based on a preemptive, time-slice basis, we can't reproduce the mouse double-clicking behavior that you described. In any case, here is the scenario of the mouse motion:
If the file is UNLOCKED and the mouse is pointed and single-clicked on the name of that file under the icon, the mouse symbol
SHOULD change from arrow pointer to I-beam which lets you change the name of the file (the name is then highlighted). Then, when the mouse is double-clicked on the name, the word in the name is selected and allowed to change that word. When the mouse pointer leaves the name area, it becomes an arrow point.
If the file is LOCKED and the mouse is pointed and a single-clicked on the name of that file under the icon, the mouse symbol remains an
arrow pointer which means the name of file can't be changed, because it was LOCKED. In this case, if the mouse is double-clicked on the name of the locked file, it will launch the program.
Check if the file to be renamed is LOCKED or UNLOCKED.